Questions tagged [real-analysis]
For questions about real analysis, a branch of mathematics dealing with limits, convergence of sequences, construction of the real numbers, least upper bound property and related analysis topics, such as continuity, differentiation, and integration through the Fundamental Theorem of Calculus.
90,248 questions
0
votes
1answer
27 views
Proving a limit converges to -3 using definition of convergence.
so I have the problem $\lim\limits_{n \to oo} \frac{2-3n^2}{n^2+2n+1}$. I have to prove this using the epsilon definition. So I know the limit equals -3. So I do
|$\frac{2-3n^2}{n^2+2n+1}$ + 3 | < $...
1
vote
0answers
20 views
A problem from Folland: constructing Haar measure from Lebesgue measure
The following is a result from A Course in Abstract Harmonic Analysis by G.B. Folland. It also appears as an exercise in the real analysis text by the same author. The context is Haar measure on a ...
0
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0answers
25 views
Prove that any bounded subset of $\mathbb R^2$ is totally bounded.
(Note I am seeking a proof before the development of the definition of compact sets; preferably one that finds an open cover of the subset.)
My attempt: I tried to connect it to a proof developed ...
2
votes
1answer
33 views
How to show that $f(x)$ is not continuous using open sets?
I am practicing topology with a book, and I'm trying to understand the following definition.
A map $f$: $\mathbb{R} \rightarrow \mathbb{R}$ is said to be continuous if for every open subset $U \...
0
votes
0answers
31 views
Prove by constructing bijective functions that the following sets $A$ and $B$ have the same cardinality. [on hold]
Prove by constructing bijective functions that the following sets $A$ and $B$ have the same cardinality:
Let $A=\mathbb{N}$, $B=\{\{x_n\}^{\infty}_{n=1}: x_n \in \mathbb{N}\cup \{0\} \hspace{0.5cm} \...
1
vote
0answers
17 views
Cantor-Like Sets
I'm working on a problem from Stein and Shakarchi's Real Analysis, and it asks to construct a measurable subset $E\subset [0,1]$ such that for any non-empty sub-interval $I$ in $[0,1]$, both $E\cap I$ ...
0
votes
0answers
12 views
Minorization Condition for a Truncated Normal Distribution
Suppose $$t_1, \ldots, t_n | \beta, \sigma^2, y \sim \text{TN}(\lambda(y_i-x_i^T\beta),\sigma^2;0;\infty),$$ where $\lambda \in \mathbb{R},$ and $x_1,\ldots, x_n$ and $y_1,\ldots,y_n$ come from the ...
1
vote
1answer
52 views
Simple proof that $\sup\{b^t : t \in \mathbb{Q} \text{ & }t≤x\} =\sup\{b^t : t \in \mathbb{Q}\text{ & }t<x\}$
Fix $b>1$. Let $B(x) = \{b^t : t \in \mathbb{Q}\text{ & }t≤x\}$ and let $B'(x) = \{b^t : t \in \mathbb{Q}\text{ & }t<x\}$.
Show that $\sup B(x) = \sup B'(x)$. It is quite easy to show ...
3
votes
4answers
53 views
Show that F vanishes at infinity.
Suppose $1 ≤ p < ∞, f ∈ L^p(R)$, and
$F(x) = \int_{x}^{x+1} f(t) dm(t)$
Prove that F vanishes at infinity.
We know that $\int_R |f|^p < \infty$, then, of course, for any $x, F(x)< \int_x^{...
6
votes
0answers
54 views
If $X$ is a complex Banach space, is the set $T \in L(X)$ with finite dimensional kernel dense?
If $X$ is a complex Banach space, is the set $T \in L(X)$ with finite dimensional kernel dense? Here $L(X)$ is the set of bounded linear operators from $X$ to itself equipped with the norm topology.
...
0
votes
0answers
31 views
Cantor's Theorem proof
I'm having problems understanding the Cantor's theorem proof. Can't we just assume $f$ is surjective, say the cardinality of $P(A)$ is greater than $A$ so by the pigeonhole principle there must be an ...
1
vote
2answers
35 views
Proving that $y_n$ converges given $y_{2n}, y_{2n+1}$ converges
Suppose we have $\mathrm{lim}_{n \rightarrow \infty} y_{2n} = \mathrm{lim}_{n \rightarrow \infty} y_{2n+1} = M \in \mathbb R$. I'm trying to prove from this that $\mathrm{lim}_{n \rightarrow \infty} ...
7
votes
2answers
58 views
Proving a specific set in $\mathbb R^2$ is closed.
I am trying to prove that the set $A=\{(x,y)|x^2\leq y\}$ is closed in $\mathbb R^2$. I wrote a proof, but I think the end is wrong. My proof is:
Consider the set $A=\{(x,y)|x^2\leq y\}$ in $\mathbb ...
0
votes
4answers
68 views
When are we not allowed to replaced $\sin{x} $ , $\tan{x}$… by $x$ in a limit where $x\to 0$?
In what situation, can we not replace for example $e^x$ with $x$ when when $x\to 0$, in a limit.
Sorry if the question is extremely vague , English is not my native language. Thank you in advance
0
votes
3answers
90 views
Can someone help me solve this limit? [on hold]
Can someone help me solve this limit? Thank you in advance.
$$\lim_{x\to 0} \bigg( \frac{1+\tan{x}}{1+\sin{x}} \bigg)^{\frac{1}{\sin{x}}}$$
If possible without L'Hospital, the exercise gives more ...
0
votes
0answers
20 views
If $f:[a,b]\to\mathbb{R}^2$ ($n>1$) is a continuous rectifiable path, then m(f([a,b]))=0 [duplicate]
If $f:[a,b]\to\mathbb{R}^2$ is a continuous rectifiable path, prove that the measure of $f([a,b])$ is null.
I've tried to use the fact that $f$ is uniform continuous, but I only got that
$m(f([a,b]...
0
votes
0answers
29 views
The set of bounded Linear operators $B(V)$. Is $B(V)$ bounded?
I have a question, regarding the set of bounded linear operators. I was wondering is that set itself bounded? Furthermore if it is not bounded then what restrictions can we put onto the vector space $...
0
votes
0answers
16 views
Understanding the proof of inversion formula for density using characteristic function
The formula is:
$f(x) = \frac{1}{2\pi}\int_{-\infty}^{\infty}e^{-i\lambda x}\hat{f}(\lambda)d\lambda$
where $\hat{f}$ is the characteristic function, $f$ is continuous bounded on
$R$ and both $f,...
3
votes
2answers
57 views
Integrable function of which the antiderivative is not equal to the integral
Is there a Riemann-integrable function $f: [a,b]\to\mathbb{R}$ such that $f$ has an antiderivative which is not equal to $t \mapsto \int_a^t f(x)dx +c$ for any $c\in\mathbb{R}$?
1
vote
1answer
50 views
Hardy-Littlewood Inequality for Sobolev spaces
After making the mistake of applying Hardy-Littlewood-Sobolev(H-L-S) for the infinity case, I was wondering if it is possible to bound it by a Sobolev norm.
Fix dimension to be $3$. H-L-S says that ...
24
votes
7answers
4k views
Does an increasing sequence of reals converge if the difference of consecutive terms approaches zero?
If $a_n$ is a sequence such that $$a_1 \leq a_2 \leq a_3 \leq ...$$
and has the property that $\space$$a_{n+1}-a_n \longrightarrow 0$,
Then can we conclude that $a_n$ is convergent?
$$\space$$
I ...
1
vote
0answers
53 views
Continuous $f:(a,b)\to \mathbb{R}$ is convex iff $x \mapsto f(x)+\gamma x+\delta$ does not attain max for all $\gamma,~\delta\in \mathbb{R}$
Let $f:(a,b)\to \mathbb{R}$ continuous. Prove that $f$ is convex iff for all $\gamma,~\delta\in \mathbb{R}$ function
$x \mapsto f(x)+\gamma x+\delta$ does not attain its maximum value on $(a,b)$.
...
1
vote
0answers
13 views
Finitely Additive Homogeneous Translation Invariant Measure on $\mathcal{P}(\mathbb{R})$
It is known that there exists a finitely additive translation invariant measure on $\mathbb{R}$ that extends the Lebesgue measure. I.e. a function $m:\mathcal{P}(\mathbb{R}) \rightarrow [0,\infty]$ ...
1
vote
1answer
42 views
(Proof verification) Showing that the two $\limsup$ definitions are equivalent
I have been trying to prove that the two definitions of $\limsup$ are equivalent. I would appreciate it if someone could verify my attempt! Thanks in advance!
Here are the two definitions:
...
1
vote
1answer
16 views
Cover $\lambda_n$-null set with small cubes
I'm reading proof that set with $\lambda_n$ measure zero have also $\mathscr{H}^n$ (Hausdorff) measure zero. And there is claim:
If $n\geq 2$, for every given $\epsilon,\delta > 0$ and Borel $\...
1
vote
1answer
47 views
A difficult limit -> problem in getting out of the form $\infty \times 0$
Let $(x_n)$ be a real sequence which converges to $l$. Moreover we have : $\mid x_n -l\mid \leq C \mid x_{n-1}-l\mid, C \in ]0,1[$ and : $\lim_{n \to \infty} \frac{x_{n+1}-l}{x_n-l} = k \in ]0,1[$. ...
0
votes
1answer
30 views
Convergence of certain integrals to $f(t)/2$
Let $f \in {L^1}\left( {0,T} \right)$. Prove that for a.e $t \in \left( {0,T} \right)$ we have
$$\mathop {\lim }\limits_{n \to + \infty } n\int\limits_t^{t + \frac{1}{n}} {\left[ {1 - n\left( {s - t}...
0
votes
1answer
23 views
Bijectiveness in a neighborhood of $(0,\pi/2)$
Question:
Define $g: \mathbb{R^2}\to \mathbb{R^2}$ by $g(x,y)=(y\cos x,(x+y)\sin y)$.
Show that g maps a neighborhood of $(0,\pi/2)$ bijectively to a neighborhood of $(\pi/2,\pi/2)$.
What I did/...
-1
votes
0answers
9 views
how functional reduce to Y(x,y,y') = y(x) + e x n(x)
How this functional can be expressed as above linear function . what math concepts and topics clear this fully and meaningful mathematical explanation.
4
votes
1answer
42 views
Trying to understand a standard example of weakly but not strongly measurable function
I consider quite a standard example of a function that is weakly measurable but is not strongly measurable. Unfortunatelly, I don't fully understand it.
Let $X=l_2([0,1])$. Then $X$ equipped with a ...
0
votes
1answer
36 views
What is the function you get when you double the arguments of sin and cos in the Fourier Series of another function?
Suppose $$f(x) =\frac{a_0}{2}+\sum_{k=1}^\infty a_k\cos(kx)+b_k\sin(kx) $$
Then what is $$S(x)=\frac{a_0}{2}+\sum_{k=1}^\infty a_k\cos(2kx)+b_k\sin(2kx)$$ in terms of $f(x)$. I tried writing down ...
1
vote
0answers
44 views
Generalized Hermite interpolation
I'm trying to understand this topic, because I found it in an exercise of Zorich - Mathematical Analysis I.
So, I have $p$ points $x_1,...,x_p$ and I want to find a polynomial $H(x)$ such that:
$f(...
1
vote
1answer
20 views
second order homogeneous differential equation method
I'm trying to understand the general method to solve the second order differential equations $u''+2au'+bu=f(t)$.
The homogeneous differential equation $u''+2au'+bu=0$ can be solved looking for a ...
0
votes
2answers
21 views
Example of a function which is continuous and $1$-periodic but not differentiable
Give an example of a function $f:\Bbb R \to \Bbb R$ such that $f$ is continuous and periodic with period $1$ but not differentible
My try
Consider the function $f$ whose graph is
Actually it is a ...
0
votes
1answer
34 views
Preimage of continuous one-to-one function on connected domain is not continuous.
I know that given $B$, a compact subset of $\mathbb{R}^n$, and $f : B \to \mathbb{R}^m$, a continuous injective (one-to-one) function, $f^{-1}$ is continuous on $f(B)$. (This true).
I also know that ...
2
votes
1answer
50 views
Exercise 36 Ch 1 in Stein's Real Analysis [duplicate]
Construct a measurable set $E\subset [0,1]$ such that for any
non-empty open sub-interval $I$ in $[0,1]$, both sets $E\cap I$ and $E^c\cap I$ have positive measure.
[Stein's Hint: For the first ...
-4
votes
0answers
26 views
Discussing the convergence of a sequence $x_n$ [on hold]
If $k, x_1$ are positive, and $x_{n+1}$ = $\sqrt {k + x_n},$ discuss the convergence of the sequence $x_n$, according to whether $x_1$ is less than or greater than $\alpha$, the positive root of $x^2 =...
2
votes
1answer
22 views
Extension of domain of contactive semigroups, and why contractivity is important?
I am reading a paper, where I think the authors use this fact that the contractivity of a semigroup on a space of initial values, imeadiatly implies that we can extend the domain of semigroup to this ...
1
vote
1answer
42 views
I am stuck somewhere in this analysis problem
Consider a real valued continuous function f on [0,1] such that f is differentiable on (0,1) and f(0)=f(1)=0. Does there exist some c in (0,1) where f(c)=f'(c)?
It seems like the answer is yes. I am ...
3
votes
0answers
46 views
Proof that continuous curve in $\mathbb{R}^2$ has Lebesgue measure zero
Suppose $\Gamma$ is a curve $y = f(x)$ in $\mathbb{R}^2$, where $f$ is continuous. Show that $m(\Gamma)=0.$
[Hint: Cover $\Gamma$ by rectangles, using the uniform continuity of $f$.]
My attempt:
...
3
votes
1answer
101 views
Prove that inequality $\sqrt{2\sqrt{4\sqrt{8…\sqrt{2^n}}}} \leqslant n+1$
Let $n$ be the integer. Prove that $$\sqrt{2\sqrt{4\sqrt{8....\sqrt{2^n}}}} \leqslant n+1$$
SOURCE: BANGLADESH MATH OLYMPIAD
I am a new beginner at the infinite radical and sequence. I don't know ...
0
votes
1answer
45 views
An analytic doubt from Riemann zeta function-Titchmarsh
Doubt form the Book Theory of Riemann Zeta Function by Titchmarsh
I've problem on Theorem $5.8$ in the equation $5.8.2$. When $k=l$ the from Lemma $5.7$ we get, $$ \sum_{n=N+1}^bn^{-it}=O\left(N^{1-1/...
1
vote
1answer
41 views
Proving $\lim_{x\to x_{0}} f(x) = 0$ implies $\lim_{x\to x_{0}} f(x)g(x) = 0$ in $\mathbb{R}^{n}$
Let $A$ be a subset of $\mathbb{R}^{n}$ and let $x_{\star} \in \mathbb{R}^{n}$ Also, suppose $g : A \rightarrow \mathbb{R}$ is bounded; that is, there is a real number $c$ so that $|g(x)| \leq c$ for ...
1
vote
0answers
24 views
Every uncountable closed set contains a perfect subset
Every uncountable closed set contains a perfect subset.
Does my attempt look fine or contain logical flaws/gaps? Any suggestion is greatly appreciated. Thank you for your help!
My attempt:
For $A \...
0
votes
0answers
28 views
Alternative proofs for Leibniz Rule
Let $f: [a,b] \times [c,d] \to \mathbb{R}$ be a continuous function with $U \subset \mathbb{R}^{n}$ and $\frac{\partial f}{\partial y}$ continuous. Take $F(x,y) = \int_{a}^{b}f(x,y)dx$ and show that
...
0
votes
1answer
26 views
Does there exist an open subset $A \subset [0,1]$ such that $m_*(A)\neq m_*(\bar{A})$?
Does there exist an open subset $A \subset [0,1]$ such that $m_*(A)\neq m_*(\bar{A})$?
I was thinking we could approximate any set from inside by a closed set . This need not true from outside.
So ...
1
vote
1answer
13 views
Consider the set $l^\infty$ of all bounded sequences of real numbers. Show that $d$ is well-defined for all pairs of sequences in $l^\infty$
$l^\infty = d(\bf{x},y) = \sup_{n > 0} |x_n - y_n|$
I am having a hard time figuring out how to show it is well-defined. We are to show $d(x,y)$ is always defined. I know that well-defined is ...
1
vote
1answer
28 views
How to show that the space of probability measures on $\mathbb{R}$ is separable under Lévy metric
The Lévy metric between distribution functions $F$ and $G$ is given by:
$$\rho(F,G) = \inf\{\epsilon : F(x-\epsilon)-\epsilon\leq G(x)\leq F(x+\epsilon)+\epsilon\}.$$
Another way to write this is: ...
1
vote
0answers
40 views
No continuous function $f$ on $\mathbb{R}$ such that $f(x)=1_{[0,1]}(x)$ almost everywhere
Let $1_{[0,1]}$ be the characteristic function of $[0,1]$. Show that there is no everywhere continuous function $f$ on $\mathbb{R}$ such that
$$f(x)=1_{[0,1]}(x)\,\,\,\,\,\,\,\,\,\,\,\,\,\text{...
4
votes
0answers
26 views
When does a convex sequence $S_n$ satisfy $S_{a+b+c} - (S_{a+b}+S_{a+c}+S_{b+c})+S_a+S_b+S_c\geq 0$
Let $S_n$ be a sequence of integers with $n\geq 0$ such that $S_0=S_1=S_2=0$ (also assume $S_n$ to not be completely zero) and
$$
S_{n-1}+S_{n+1}\geq 2S_n
$$
i.e. $S_n$ is convex. Now consider the ...
